The inclusion $\mathcal C\subseteq\mathcal D$ is a <structure embedding> between <first-order models> of the same <complete theory>, but it is not an <elementary embedding>, as the existential partner <first-order formula> at $u$ shows. Therefore \b[the <first-order theory> is not model-complete]. The facts $\mathcal B\preceq\mathcal C$ and $\mathcal B\preceq\mathcal D$ do not make the intervening inclusion elementary.

Solution

Past exam of the model theory course of the University of Oxford / 2011 / 2 / c / iv / Solution

 Solution (source code)